Relaxed Graceful Labellings of Trees

A graph $G$ on $m$ edges is considered graceful if there is a labelling $f$ of the vertices of $G$ with distinct integers in the set $\{0,1,\dots,m\}$ such that the induced edge labelling $g$ defined by $g(uv)=|f(u)-f(v)|$ is a bijection to $\{1,\dots,m\}$. We here consider some relaxations of these conditions as applied to tree labellings: 1. Edge-relaxed graceful labellings, in which repeated edge labels are allowed, 2. Range-relaxed graceful labellings, in which the upper bound $m'$ is allowed to go higher than the number of edges, and 3. Vertex-relaxed graceful labellings, in which repeated vertex labels are allowed. The first of these had been looked at by Rosa and Širáň (1995). Here some linear bounds in the relevant metrics are given for range-relaxed and vertex-relaxed graceful labellings.